Prime Factors and Index Notation

What are Prime Factors?

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A prime factor is any prime number that divides evenly into another number (without leaving any remainder); and thus, forming a unique set of prime numbers that will multiply together to give the original number.

Recall that a prime number is any whole number which is greater than 1; and divisible by 1 and itself only.

 

In order to find the prime factors of a number, we can use the division method by repeatedly dividing the number by the smallest possible prime number (2, 3, 5, 7, 9 and so on) until we're left with 1. During this division, we'll collect all the prime divisors.

Please read the introduction to prime numbers here.

 

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Steps Taken to Find Prime Factors

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Below are the steps to be taken when finding the prime factors through the division method:

• Step 1: Divide the number (in the question) by the smallest prime number (which is a divisor that often begins by 2), without leaving a remainder.

• Step 2: Take the result (quotient) in step 1 and divide it by the smallest prime number (a divisor) that divides it evenly.

• Step 3: Keep dividing the result (quotient) by the smallest prime number that divides it evenly. Continue this process and stop when your result becomes 1.

• Step 4: Collect the prime factors. These are all the prime numbers which you used as divisors.

Note: The value '1' is not included as part of your prime factors (because 1 isn't a prime factor).

Please read on highest common factor (HCF) here.

 

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Worked Examples on Prime Factors

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Example 1

• Find the prime factors of 10, their sum and product.

Solution 

2	10
5	5
	1

 

• The prime factor of 10 is 2 and 5

• The sum of the prime factors of 10 = 2 + 5 = 7

• The product of the prime factors: 2 × 5 = 10

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Example 2

• Find the prime factors of 16, their sum and product.

Solution

2	16
2	8
2	4
2	2
	1

 

• The prime factor of 16 is 2

• The sum of the prime factors of 16  = 2 + 2 + 2 + 2 = 8

• The product of the prime factors: 2 × 2 × 2 × 2 = 16

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Example 3

• Find the prime factors of 36, their sum and product.

Solution

2	36
2	18
3	9
3	3
	1

 

• The prime factors of 36 are 2 and 3

• The sum of the prime factors of 36 = 2 + 2 + 3 + 3 = 10

• The product of the prime factors: 2 × 2 × 3  × 3 = 36

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Example 4

• Find the prime factors of 108, their sum and product.

Solution 

2	108
2	54
3	27
3	9
3	3
	1

 

• The prime factors of 108 are 2 and 3

• The sum of the prime factors of 108 = 2 + 2 + 3 + 3 + 3 = 13

• The product of the prime factors: 2 × 2 × 3 × 3 × 3 = 108

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Example 5

• Find the prime factors of 150, their sum and product.

Solution

2	150
3	75
5	25
5	5
	1

 

• The prime factors of 150 are 2, 3 and 5

• The sum of the prime factors of 150 = 2 + 3 + 5 + 5 = 15

• The product of the prime factors: 2 × 3 × 5 × 5 = 150

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Example 6

• Find the prime factors of 180, their sum and product.

Solution 

2	180
2	90
3	45
3	15
5	5
	1

 

• The prime factor of of 180 are 2, 3 and 5

• The sum of the prime factors of 180 = 2 + 2 + 3 + 3 + 5 = 15

• The product of the prime factors: 2 × 2 × 3 × 3 × 5 = 180

Please read on the factors of numbers here.

 

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Index Notation of Prime Factors 

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The index notation of prime factors is an act of expressing a number as a product of its unique prime factors raised to the power of the number of their occurrence (that is, their exponent).

 

Below are steps for writing index notation:

• Step 1: Determine the base number (or variable) that is being multiplied by itself (for example '2 × 2 × 2' or z × z × z × z')

• Step 2: Count how many times the base number is multiplied by itself. This count becomes the exponent (or index). In the above instance, '2' is multiplied by itself three times while 'z' is multiplied by itself four times.

• Step 3: Write the base with the index (or exponent) as a superscript. In our example above, it becomes 2³ and z⁴ respectively; and that's our index notation.

You can read on expressing one quantity as a factor of another here

 

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Worked Examples on Index Notation

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• Express the following numbers in index form:

Example 1

• 7 x 7 x 7

Solution 

• The index notation is 7³

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Example 2

• 2 × 2 × 2 × 3 × 3 × 5 × 5 × 7 × 11 × 11

Solution 

• The index notation is 2³ × 3² × 5² × 7 × 11²

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• Express each of the following as a product of its prime factors leaving your answer in index notation.

Example 3

• 50

Solution 

2	50
5	25
5	5
	1

 

• Product of prime factor = 2 × 5 × 5

• Index notation = 2 × 5²

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Example 4

• 675

Solution 

3	675
3	225
3	75
5	25
5	5
	1

 

• Product of prime factor = 3 × 3 × 3 × 5 × 5

• Index notation = 3³ × 5²

You can read on the types of angles based on degree measurement here.

 

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Assignment on Prime Factors and Index Notation

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Find the prime factors and index notation of the following numbers:

• 200

• 508

• 305

• 2024

• 130

• 940

• 330

• 489

• 100

• 1250

• 530

• 670

• 750

• 880

• 910

Please read on mathematics week 3 topic for jss1 first termm here.

Please read on mathematics scheme of work for jss1 first term here.